<h2>题目编号 : 128</h2>
<div style="color:#666;font-size:80%;">29 September 2006</div><br />
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<p>A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at &quot;12 o'clock&quot; and numbering the tiles 2 to 7 in an anti-clockwise direction.</p>
<p>New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first three rings.</p>
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<img src='project/images/p_128.gif' width='400' height='431' alt='' />
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<p>By finding the difference between tile <i>n</i> and each its six neighbours we shall define PD(<i>n</i>) to be the number of those differences which are prime.</p>
<p>For example, working clockwise around tile 8 the differences are 12, 29, 11, 6, 1, and 13. So PD(8) = 3.</p>
<p>In the same way, the differences around tile 17 are 1, 17, 16, 1, 11, and 10, hence PD(17) = 2.</p>
<p>It can be shown that the maximum value of PD(<i>n</i>) is 3.</p>
<p>If all of the tiles for which PD(<i>n</i>) = 3 are listed in ascending order to form a sequence, the 10th tile would be 271.</p>
<p>Find the 2000th tile in this sequence.</p>

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